# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008 QR
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• ### 2022-05-01

#### The exceptional set of one prime square and five prime cubes

Yuhui Liu

Abstract : For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.

• ### 2022-01-01

#### Gorenstein sequences of high socle degrees

Jung Pil Park, Yong-Su Shin

Abstract : In , the authors showed that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$ and $h_i\le h_1$ is a Gorenstein sequence, then $h_1=h_i$ for every $1\le i\le e-1$ and $e\ge 6$. In this paper, we show that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $h_2=4e-3$, and $h_i\le h_2$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 7$. We also propose an open question that if  an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $4e-3<h_2\le {(h_1)_{(1)}}| ^{+1}_{+1}$, and $h_2\le h_i$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 6$.

• ### 2022-03-01

#### Ring isomorphisms between closed strings via homological mirror symmetry

Sangwook Lee

Abstract : We investigate how closed string mirror symmetry is related to homological mirror symmetry, under the presence of an explicit geometric mirror functor.

• ### 2022-01-01

#### The interior gradient estimate for a class of mixed Hessian curvature equations

Jundong Zhou

Abstract : In this paper, we are concerned with a class of mixed Hessian curvature equations with non-degeneration. By using the maximum principle and constructing an auxiliary function, we obtain the interior gradient estimate of $(k-1)$-admissible solutions.

• ### 2022-03-01

#### Goldbach-Linnik type problems with unequal powers of primes

Li Zhu

Abstract : It is proved that every sufficiently large even integer can be represented as a sum of two squares of primes, two cubes of primes, two fourth powers of primes and 17 powers of 2.

• ### 2022-03-01

#### Generalized Killing structure Jacobi operator for real hypersurfaces in complex hyperbolic two-plane Grassmannians

Hyunjin Lee, Young Jin Suh, Changhwa Woo

Abstract : In this paper, first we introduce a new notion of generalized Killing structure Jacobi operator for a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with generalized Killing structure Jacobi operator.

• ### 2022-05-01

#### Zeros of new Bergman kernels

Noureddine Ghiloufi, Safa Snoun

Abstract : In this paper we determine explicitly the kernels $\mathbb K_{\alpha,\beta}$ associated with new Bergman spaces $\mathcal A_{\alpha,\beta}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $\alpha\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{\alpha,\beta}$. Some numerical results are given throughout the paper.

• ### 2022-01-01

#### Grothendieck group for sequences

Xuan Yu

Abstract : For any category with a distinguished collection of sequences, such as $n$-exangulated category, category of N-complexes and category of precomplexes, we consider its Grothendieck group and similar results of Bergh-Thaule for $n$-angulated categories  are proven. A classification result of dense complete subcategories is given and we give a formal definition of K-groups for these categories following Grayson's algebraic approach of K-theory for exact categories .

• ### 2022-03-01

#### Cohomology of torsion and completion of $N$-complexes

Pengju Ma, Xiaoyan Yang

Abstract : We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes using the $\check{\mathrm{C}}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the categories of cohomologically $\mathfrak{a}$-torsion $N$-complexes and cohomologically $\mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative Noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $\otimes$ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.

• ### 2022-03-01

#### A deep learning algorithm for optimal investment strategies under Merton's framework

Daeyung Gim, Hyungbin Park

Abstract : This paper treats Merton's classical portfolio optimization problem for a market participant who invests in safe assets and risky assets to maximize the expected utility. When the state process is a $d$-dimensional Markov diffusion, this problem is transformed into a problem of solving a Hamilton--Jacobi--Bellman (HJB) equation. The main purpose of this paper is to solve this HJB equation by a deep learning algorithm: the deep Galerkin method, first suggested by J. Sirignano and K. Spiliopoulos. We then apply the algorithm to get the solution to the HJB equation and compare with the result from the finite difference method. ## Current Issue

• ### Katok-Hasselblatt-kinematic expansive flows

Hien Minh Huynh

J. Korean Math. Soc. 2022; 59(1): 151-170
https://doi.org/10.4134/JKMS.j210258

• ### Global nonexistence for the wave equation with boundary variable exponent nonlinearities

Tae Gab Ha, Sun-Hye Park

J. Korean Math. Soc. 2022; 59(1): 205-216
https://doi.org/10.4134/JKMS.j210364

• ### Hardy type estimates for Riesz transforms associated with Schr\"{o}dinger operators on the Heisenberg group

Chunfang Gao

J. Korean Math. Soc. 2022; 59(2): 235-254
https://doi.org/10.4134/JKMS.j200484

• ### Margin-based generalization for classifications with input noise

Hi Jun Choe, Hayeong Koh, Jimin Lee

J. Korean Math. Soc. 2022; 59(2): 217-233
https://doi.org/10.4134/JKMS.j200406

• ### Regularity of the generalized Poisson operator

Pengtao Li, Zhiyong Wang, Kai Zhao

J. Korean Math. Soc. 2022; 59(1): 129-150
https://doi.org/10.4134/JKMS.j210224

• ### Practical FHE parameters against lattice attacks

Jung Hee Cheon, Yongha Son, Donggeon Yhee

J. Korean Math. Soc. 2022; 59(1): 35-51
https://doi.org/10.4134/JKMS.j200650

• ### Global nonexistence for the wave equation with boundary variable exponent nonlinearities

Tae Gab Ha, Sun-Hye Park

J. Korean Math. Soc. 2022; 59(1): 205-216
https://doi.org/10.4134/JKMS.j210364

• ### Construction for self-orthogonal codes over a certain non-chain Frobenius ring

Boran Kim

J. Korean Math. Soc. 2022; 59(1): 193-204
https://doi.org/10.4134/JKMS.j210357